Quiver Poisson algebras

Inner Poisson algebras on a given associative algebra are introduced and characterized, which gives a way of constructing non-commutative Poisson structures. Applying these to the finite-dimensional path algebras , together with the decomposition into indecomposable Lie ideals of the standard Poisson structure on , we classify all the inner Poisson structures on , which turn out to be the piecewise standard Poisson algebras. We also determine all the finite quivers without oriented cycles such that admits outer Poisson structures: these are exactly the finite quivers without oriented cycles such that there exist two non-trivial paths α and β lying in a reduced closed walk, which cannot be connected by a sequence of non-trivial paths.